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1 Name Final Examination December 15, 2010 Circle your TA and principle recitation instructor so that we can more easily identify with whom you have studied: Martin Couturier Kenny Donahue Gleb Kuznetsov Kendra Pugh Mark Seifter Yuan Shen Robert Berwick Randall Davis Lisa Fisher Indicate the approximate percent of the lectures, mega recitations, recitations, and tutorials you have attended so that we can better gauge their correlation with quiz and final performance and with attendance after OCW video goes on line. Your answers have no effect on your grade. Percent attended Lectures Recitations Megas Tutorials Quiz Score Grader Q1 Q2 Q3 Q4 Q5 There are 48 pages in this final examination, including this one. In addition, tearoff sheets are provided at the end with duplicate drawings and data. As always, open book, open notes, open just about everything. 1

2 Quiz 1, Problem 1, Rule Systems (50 points) Kenny has designed two suits for the Soldier Design Competition, and he and Martin like to grapple in the prototypes on Kresge Oval. Kendra insists the suits qualify as deadly weapons and Kenny should give them to her for safekeeping. Kenny and Martin insist that they are examples of an enhanced prosthesis and that they should be able to keep them The TAs decide to use Rule-Based Systems to resolve their dispute. Rules: P0 P1 P2 P3 P4 IF (AND ('(?x) is a Crazy Physicist', '(?x) is an Engineer') THEN ('(?x) builds a Weaponized Suit') ) IF ('(?y)'s (?x) is Cardinal Red' THEN ('(?y)'s (?x) is not US Govt. Property') ) IF (OR (AND ('(?y) is an Engineer', '(?y)'s (?x) is Really Heavy'), '(?y)'s (?x) is stolen by the Air Force') THEN ('(?y)'s (?x) is a Deadly Weapon') ) IF (OR ('(?y) is not evil', '(?y) is a Roboticist') THEN ('(?y)'s suit research is not Evil') ) IF (AND ('(?y)'s (?x) research is not Evil', '(?y)'s (?x) is not US Govt. Property') THEN ('(?y)'s (?x) is an Enhanced Prosthesis') ) Assertions: A0: (Kenny is a Roboticist) A1: (Martin is an Engineer) A2: (Kenny's suit is Cardinal Red) A3: (Martin's suit is Really Heavy) 2

3 Part A: Backward Chaining (30 points) Make the following assumptions about backward chaining: The backward chainer tries to find a matching assertion in the list of assertions. If no matching assertion is found, the backward chainer tries to find a rule with a matching consequent. In case none are found, then the backward chainer assumes the hypothesis is false. The backward chainer never alters the list of assertions; it never derives the same result twice. Rules are tried in the order they appear. Antecedents are tried in the order they appear. Simulate backward chaining with the hypothesis Kenny's suit is an enhanced prosthesis Write all the hypotheses the backward chainer looks for in the database in the order that the hypotheses are looked for. The table has more lines than you need. We recommend that you use the space provided on the next page to draw the goal tree that would be created by backward chaining from this hypothesis. The goal tree will help us to assign partial credit in the event you have mistakes on the list. 1 Kenny's suit is an enhanced prosthesis

4 Draw Goal Tree Here for Partial Credit 4

5 Part B: Forward Chaining (20 points) Let's say, instead, our assertions list looked like this: A0: Gleb is an Engineer A1: Gleb's laptop is Really Heavy A2: Gleb's suit is Really Heavy B1 (4 points) CIRCLE any and all rules that match in the first iteration of forward chaining B2 (4 points) P0 P1 P2 P3 P4 What assertion(s) are added or deleted from the database, as a consequence of this iteration? B3 (4 points) CIRCLE any and all rules that match in the second iteration of forward chaining P0 P1 P2 P3 P4 B4 (4 points) What assertion(s) are added or deleted from the database, as a consequence of this iteration? B5 (4 points) You take the same assertions as at the beginning of problem B, above, and re-order them: A0: Gleb is an Engineer A1: Gleb's suit is Really Heavy A2: Gleb's laptop is Really Heavy Now, you start over, and run forward chaining from the beginning, until no new assertions are added to or deleted from the database. Is Gleb's laptop a Deadly Weapon? 5

6 Quiz 1, Problem 2, Search (50 points) As you get close to graduating MIT, you decide to do some career planning. You create a graph of your options where the start node is M = MIT and your goal node is R = Retire, with a bunch of options in between. Your graph includes edge distances that represent, roughly, the cost of transition between these careers (don't think too hard about what this means). You also have heuristic node-to-goal distances which represent your preconceptions about how many more years you have to work until you retire. For example, you think it will take 25 years to go from MIT (M) to retirement (R), 30 years from Grad School (B), but only 2 years from Entrepreneur (E). A = Wall Street B = Grad School C = Professor D = Government E = Entrepreneur Part A: Basic search (25 points) In all search problems, use alphabetical order to break ties when deciding the priority to use for extending nodes. 6

7 A1 (3 points) Assume you want to retire after doing the least number of different jobs. Of all the basic search algorithms you learned about(that is, excluding branch and bound and A*) which one should you apply to the graph in order to find a path, with the least search effort, that has the minimum number of nodes from M to R? A2 Basic Search Chosen Above (7 points) Perform the search you wrote down in A1 (with an Extended List). Draw the search tree and give the final path. Tree: Path: 7

8 A3 Beam Search with w=2 (15 points) Now you are interested in finding a path and the associated distance. Try a Beam Search with a width w=2, with an extended list. As before, you are looking for a path from M to R. Use the preconceptions heuristic distances indicated in parentheses at each node. Tree: Path, if any: Extended nodes in order extended: 8

9 Part B: Advanced Search (25 points) B1 Branch and Bound with Extended List (15 points) Use Branch and Bound search with an Extended List to find a path from M to R, as well as the extended node list. Use this space to draw the corresponding tree and show your work. Tree: Path: Extended nodes in order extended: 9

10 B2 Thinking about Search (9 points) Concisely explain why Branch and Bound with Extended List yields a different result than Beam Search in this problem. What can we say about the path found by Branch and Bound with Extended List? (We're looking for a fairly strong statement here.) Is there an algorithm that guarantees the same answer as Branch and Bound for the graph in this problem, but can find the answer with fewer extended paths. If Yes, what is that algorithm? If No, explain why not. 10

11 B3 Permissible Heuristics (6 points) Suppose you are asked to find the shortest path from H to G in the graphs below. For both of the graphs explain why the heuristic values shown are not valid for A*. Note the differences in the graphs at nodes F and E. Reason(s): Reason(s): 11

12 Quiz 2, Problem 1, Games (50 points) Part A: Basics(15 points) A1 Plain Old Minimax(7 points) Perform Minimax on this tree. Write the minimax value associated with each node in the box below, next to its corresponding node letter. A= B= C= D= E= F= G= A2 Tree Rotations (8 points) Using the minimax calculations from part A1, without performing any alpha-beta calculation, rotate the children of each node in the above tree at every level to ensure maximum alpha-beta pruning. - Fill in the nodes with the letter of the corresponding node. Draw the new edges 12

13 Part B: Alpha Beta (35 points) B1: Straight-forward Alpha Beta(15 points) Perform Alpha Beta search on the following tree. - Indicate pruning by striking through the appropriate edge(s). - Mark your steps for partial credit. - Fill in the number of static evaluations. - List the leaf nodes in the order that they are statically evaluated. Indicate in Next Move which of B or C you would go to from A and in Moving Towards which node in the bottom row you are heading toward. # of evaluations: List: Next Move: Moving towards: 13

14 B2: Preset Alpha-Beta (15 points) Perform alpha-beta search, using initial values of alpha = 5 and beta = 8. - Indicate pruning by striking through the appropriate edge(s). - Mark your steps for partial credit. - Fill in the number of static evaluations. - List the leaf nodes in the order that they are statically evaluated. Indicate in Next Move which of B or C you would go to from A and in Moving Towards which node in the bottom row you are heading toward. # of evaluations: List: Next Move: Moving towards: B3: Alpha-Beta Properties (5 points) If you were able to maximally prune a tree while performing Alpha-Beta search, approaximately how many static evaluations would you end up doing for a tree of depth d and branching factor b? 14

15 Quiz 2, Problem 2, Constraint Propagation (50 points) After taking 6.034, your decide to offer your services to incoming freshman to help them set up their course schedules. One particular freshman comes to you with four classes as well as an availability schedule (grayed out boxes represent reserved times). Course Lecture Times Offered Recitation Times Offered MWF 11,12 TR 10,11,12, MWF 12, 1 TR 11,1,2 8.01T MWF 10, 11, 12, 1, 2, 3 NONE 21F.301 MTWRF 10, 11 NONE Time MWF TR For easier bookkeeping you adopt the folowing naming convention (L = Lecture, R = Recitation, # = course number): Lecture Recitation 8.01T Lecture Lecture Recitation 21F.301 Lecture L3 R3 L8 L18 R18 L21 MWF10 MWF11 TR10 TR11 10M 11M 10T 11T You also devise this set of constraints for yourself: (1) Each class must be assigned to exactly one timeslot (2) Each timeslot can be assigned to a maximum of one class (3) No classes can be scheduled during the grayed out time periods (4) The TR selection for 21F.301 must occur at the same time as the MWF selection. 15

16 Part A: Picking a Representation (8 points) In order to fill in this schedule, you decide to set it up as a CSP using meeting times as variables and courses as the values of in the domains. After filling in the domain table, this is what you see: Variable 10M 11M 12M 1M 2M 3M 10T 11T 12T 1T 2T 3T Domain L8, L21 L3, L8, L21 L3, L18, L8 L8 R3, L21 R3 R18 What is wrong with the way that this problem is set up and why? 16

18 B2 (16 Points) Run the DFS with forward checking only on your defined variables and the reduced domains you found in Part B1 by applying Constraint(3). L8 L3 R3 L21 (MWF) L21 (TR) L18 R18 18

19 B3 (5 Points) How many times did the algorithm need to backtrack? B4 (10 Points) It occurs to you that you may be able to accelerate the process of finding a solution if you were to perform domain reduction with propagation through singletons before running the DFS. Fill in your updated domain table with the results of your computations. Variable Domain L8 L3 R3 L21 (MWF) L21 (TR) L18 R18 19

21 Quiz 3, Problem 1 KNN and ID Trees (50 points) After receiving yet another Dear sir or madam.. , you decide to construct a spam filter. Part A: Nearest Neighbors (25 points) For your first attempt you decide to try using a k Nearest Neighbors model. You decide to classify spam according to 2 features: word count and occurrences of the words sir or madam. A1 (10 points) Draw the decision boundary for 1-nearest-neighbor on the above diagram of the given training data. Use the center of the faces as the positions of the training data points. 21

22 A2 (8 points) How will 1-nearest-neighbor classify an with 200 words of which 9 are the word sir? Plot this point on the graph as X? (2pts) How will 3-nearest-neighbors classify an with 600 words of which 7 are the word madam? Plot this point on the graph as Y? (3pts) How will 5-nearest-neighbors classify an with 500 words of which 25 are the word madam? Plot this on the graph as Z? (3pts) A3 (7 points) List which points yield errors when performing leave-one-out cross validation using 1-nearestneighbor classification. (3 pts) How would one go about selecting a good k to use? (4 pts) 22

23 Part B: ID Trees (25 points) Realizing nearest neighbors may not be the best tool for building a spam filter, you decide to try another classifier you learned about in 6.034: Identification Trees. B1 (8 points) It appears that the over-use of the words sir or madam seems to be a strong hint at an being spam. What is the minimum disorder and minimum-disorder decision boundary when you consider only the dimension of Sir or Madam? You can use fractions, real numbers, and logarithms in your answer. Approximate boundary: Associated Disorder: B2 (8 points) Suppose we were given the following additional information about our training set: s I, G, J, H, K and L are from Anne Hunter. One of those s might be important so you don't want to risk missing a single one so you re-label all Anne Hunter s in the training data set to be good s. You are to find the best axis-parallel test given the revised labellings of good and spam. (NOTE: Use the unlabeled graphs in the tear-off sheets section if you need it to visualize the modified data). B2.1 Which s does your new test missclassify on the modified data? (4pts) 23

24 B2.2 What is the disorder of your new test on the modified training data set? Leave your answer as a function of fractions, real numbers, and logarithms. (4pts) B3 (9 points) Soon, you decide that your life goal is no longer to be a tool for a Harvard or Sloanie startup so you decide that all s from Anne Hunter should be marked as spam. (Again, use the unlabeled graphs in the tear-off sheets if you need them). Given the revised determination of what is good and spam, draw the disorder minimizing identification tree that represents your fully trained ID-tree spam filter. You may use any horizontal and vertical classifiers in the dimensions of word count and sir or madam occurrences. Ties should be broken in order of horizontal then vertical classifiers. 24

25 Quiz 3, Problem 2, Neural Nets (50 Points) Note that this problem has three completely independent parts. Part A: Derivations (14 pts) A1. (7 pts) Using what you've learned from doing lab 5, write out the equation for dp dw CE expressed in terms of o i, d, and/or any weights and constants in the network. (o i refers to the output of any neuron in the network.) 25

26 A2. (7 pts) Write out the equation for and constants in the network. do E dw XA expressed in terms of do i dw XA, o i, and/or any weights 26

27 Part B: Letter Recognition (20 pts) You propose to use a neural network to recognize characters from a scanned page. Letters are represented binary images on a 1x1 unit grid. Assume that scaling and rotation are all done. Because you want to start with something easy, you start with the problem of recognizing a character as either possibly a T or definitely not a T. During training, each training sample consists of a random point, (x, y), along with the desired 0 or 1 value: 1 if the underlying pixel at (x, y) is part of a T; 0 if the pixel is part of a T's background. You want to find the most compact network that will correctly handle the T problem, so you decide to analytically work out the minimal network that will correctly classify a character as possibly a T or definitely not a T. Assume you decide to have the above network architecture, fill in the 7 missing weights in the table that are required to accurately classify all points in the image for T. Your weights must be integer weights or integer constraints on weights! Show your work for partial credit: Answers: W XA 0 W XC W YA W YC 0 W A W C W XB W AD W YB W BD 2 W B 2 W CD 2 W D 3 27

28 Show work here for partial credit: 28

29 Part C: Expressive Power of Neural Nets (16 pts) Circle all the functions that these networks are theoretically able to fully learn, and if your answer is No indicate the lowest possible error rate. List of Functions: X AND Y X = Y X = 0.5 AND Y = 0.5 X-Shape C1 Can be fully learned? The minimum # error if No X AND Y Yes No X = Y Yes No X = 0.5 AND Y = 0.5 Yes No X Shape Yes No How about when all initial weights are set to the same constant k? C2 Can be fully learned? The minimum # error if No X AND Y Yes No X = Y Yes No X = 0.5 AND Y = 0.5 Yes No X Shape Yes No 29

30 Quiz 4, Problem 1, Support Vector Machines (50 points) Part A: Solving SVMs (40 pts) You decided to manually work out the SVM solution to the CORRELATES-WITH function. First you reduce your problem to looking at four data points in 2D (with axes of x 1 and x 2 ). You ultimately want to come up with a classifier of the form below. More formulas are provided in tear off sheets. h x : a i y i K x, x i b 0 Your TA, Yuan suggests that you use this kernel: K u, v =u 1 v 1 u 2 v 2 u 2 u 1 v 2 v 1 A1(5 pts): Note that Φ u is a vector that is a transform of u and the kernel is the dot product, Φ u Φ v. Determine the number of dimensions in Φ u, and then write the components of Φ u in terms of u 's x 1 and x 2 components, u 1 and u 2. Explain why it is better to use Φ u rather than u. Φ u 's components: Why better: 30

31 A2(10 pts): Fill in the unshaded portion in the following table of Kernel values. K(A,A) = K(A,B) = K(A,C) = K(A,D) = K(B,A) = K(B,B) = K(B,C) = K(C,A) = K(C,B) = K(C,C) = K(D,A) = K(D,B) = K(D,C) = K(B, D) = K(C, D) = K(D, D) = A3 (10 pts): Now write out the full constraints equations you'll need to solve this SVM problem They should be in terms of α i, b, and constants. (Hint: The four data points lie on their appropriate gutters). 1 Constraint Equations A4 (5 pts): Instead of solving the system of equations for alphas, suppose the alphas were magically given to you as: α A =1/9 α B =1/9 α C =1/9 α D =1/9 b=1 Compute w Given your alphas. Hint: The number of dimensions in w is the same as the number of dimensions of Φ x w = 31

32 A5 (5 pts): What is the equation of the optimal SVM decision boundary using the answers and values from A5? h x = A6 (5 pts): What is the width of the road defined by the optimal SVM decision boundary? Width = 32

33 Part B: Kernel for k-nearest Neighbors (10 pts) A student noticed that Kernel methods can be used in other classification methods, such as k-nearest neighbors. Specifically, one could classify an unknown point by summing up the fractional votes of all data points, each weighted using a Radial Basis Kernel. The final output of an unknown point x depends on the sum of weighed votes of data points x i in the training set, computed using the function: x x i K x, x i =exp 2 s Negatively labeled data points contribute -1 times the kernel value and positively labeled training data points contribute +1 times the kernel value. You may find the graphs provided in the tear off sheets helpful. B1 (6 pts) Using this approach, as s increases to infinity, what k does this correspond to in k-nn? As s decreases to zero, what k does this approximately correspond to in k-nn? B2 (4 pts): State a reason why you would prefer to use the SVM with Radial Basis Kernel solution rather than the method of Weighted Nearest Neighbors with a Gaussian Kernel. 33

34 Quiz 4, Problem 2, Boosting (50 points) Kenny wants to recognize faces in images. He comes up with a few things that he thinks will probably work well as weak classifiers and decides to create an amalgam classifier based on his training set. Then, given an image, he should be able to classify it as a FACE or NOT FACE. When matched against faces, the GRAY part of a classifier can be either WHITE or BLACK) Classifiers: Name Image Representation A Has Hair B Has Forehead C Has Eyes D Has Nose E Has Smile F Has Ear 34

38 B2 Amalgam Classifier (5 points) What is the overall classifier after 4 rounds of boosting? What is its error rate? Part C: Miscellaneous (15 points) C1 Using Classifiers (5 points) Assume the boosting setup in Part B occurs for 100 rounds whether or not the overall classifier gets to the point where all training samples are correctly classified. Place the four classifiers in the following two bins Frequently selected: Infrequently selected: C2 More using Classifiers (5 points) Which three classifiers from A-N would you use so that you would not have to do boosting to get a perfect classifier for the samples. C3 Even more using Classifiers (5 points) Now, Suppose you are working with just two classifiers, neither of which has a zero error rate. Will boosting converge to an overall classifier that perfectly classifies all the training samples? Yes Explain: No 38

39 Quiz 5, Problem 1, Probability (50 points) Part A: Life Lessons in Probability (22 pts) Consider the following inference net developed for students who graduate from MIT: I = Had quality Instruction H = Hard working ethic T = Raw Talent S = Successful in life C = Confidence W = Took

40 A1: What is the probability that a student (W = true) had quality instruction (I = true) and became successful in life (S = true), but did not have raw talent (T = false) yet was hardworking H = true) and confident (C = true). Leave your answer unsimplified in terms of constants from the probability tables. (6pts) For A2-A3: Express your final answer in terms of expressions of probabilities that could be read off the Bayes Net. You do not need to simplify down to constants defined in the Bayes Net tables. You may use summations as necessary. A2: What is probability of success in life (S = true) given that a student has high quality instruction (I = true)? (6 pts) A3: What is the probability a student is hardworking (H = true), given that s/he was a student (W = true)? (10 pts) 40

42 Part C: Coin Tosses (16 pts) You decide to reexamine the coin toss problem using model selection. You have 2 types of coins: 1) Fair: P(heads) = 0.5 2) All-heads: P(heads) = 1.0 You have 2 possible models to describe your observed sequences of coin types and coin tosses. 1) Model 1: You have both types of coins and you draw one of them at random, with equal probability, and toss it exactly once. You repeat both the drawing and tossing 3 times total. 2) Model 2: You have both types of coins and you draw one of them at random, with equal probability, and toss it exactly 3 times. Finally, you have the following observed data. Toss 1 Toss 2 Toss 3 H H T The following questions use your knowledge of model selection to determine which is most likely. You decide to use the following criterion to weigh models: P M = 1 1 where Z is a normalization constant. Z parameters parameters is defined as the number of cell entries in the CPTs of the Bayes Net representation. C1. What is P(Model 1)? (3 pts) (Partial credit if you sketch the Models as Bayes Nets) What is P(Model 2)? (3 pts) 42

43 You've decided that the a priori model probability P(Model) to use should be uniform. P(Model 1) = P(Model 2) Under this assumption you decide to work out the most likely model, given the data, P(Model Data). C2 What is the most-likely model based on this fully observed data set: (10 pts) P(Model 1 Data)? P(Model 2 Data)? Therefore the most likely model is: (circle one) Model 1 Model 2 43

44 Quiz 5, Problem 2, Near Miss (20 points) Having missed many tutorials, lectures, and recitations, Stew is stuck on trying to figure out who are the TAs in You, who is more faithfully attending, knows who is who. Armed with your knowledge about Near Miss concept learning. You decide to build a model that will help Stew figure out who the TAs are. The following table summarizes the training data about the current staff of The Title attribute is organized as a tree, with MEng and PhDs both a type of Student. Students and Faculty are grouped under the type People-You-See-On-Campus Name TA Hair Color Title Glasses Gender Experience # Years Kendra Yes Brown MEng yes female 1 Yes Kenny Yes Brown MEng no male 1 Yes Martin Yes Black MEng no male 1 Yes Mark Yes Black PhD no male 4 Yes Mich Yes Blonde MEng yes male 10 No Gleb Yes Brown MEng no male 2 Yes Yuan Yes Black PhD yes male 3 No Lisa No Blond Professor yes female 10 No Bob No Brown Professor no male 10 No Randy No Brown Professor no male 10 No Taken

45 Fill in the table to build a model of a TA. Mark an attributes as "?" if the property has been dropped. Example Heuristics Used Model Description TAs Hair Color Title Glasses Gender Experience Taken Kendra Initial Model Brown MEng yes female 1 Yes Kenny Martin Yuan Bob Fill in the following table to build a model of a Faculty Member (FM). You may assume that Patrick, Randy, Bob, and Lisa are faculty members. Example Heuristics Used Model Description FMs Hair Color Title Glasses Gender Experience Taken Patrick Initial Model Blond Professor Yes Male 10 No Randy Bob Mich Lisa 45

47 Quiz 5, Problem 3, Big Ideas (30 points) Circle the best answer for each of the following question. There is no penalty for wrong answers, so it pays to guess in the absence of knowledge. 1 Ullman's alignment method for object recognition 1. Is an effort to use neural nets to detect faces aligned with a specified orientation 2. Relies on a presumed ability to put visual features in correspondence 3. Uses A* to calculate the transform needed to synthesize a view 4. Uses a forward chaining rule set 5. None of the above 2 Ullman's intermediate-features method for object recognition 1. Is an effort to use boosting with a classifier count not too small and not too large 2. Is an example of the Rumpelstiltskin principle 3. Is a demonstration of the power of large data sets drawn from the internet 4. Uses libraries containing samples (such as nose and mouth combinations) to recognize faces 5. None of the above 3 The SOAR architecture is best described as 1. A commitment to the strong story hypothesis 2. A commitment to rule-like information processing 3. An effort to build systems with parts that fail 4. The design philosophy that led to the Python programming language 5. None of the above 4 The Genesis architecture (Winston's research focus) is best described as, in part, as 1. A commitment to the strong story hypothesis 2. Primarily motivated by a desire to build more intelligent commercial systems 3. A commitment to rule-like information processing 4. A belief that the human species became gradually smarter over 100s of thousands of years. 5. None of the above 47

48 5 A transition frame 1. Focuses on movement along a trajectory 2. Focuses on the movement from childlike to adult thinking 3. Focuses on a small vocabulary of state changes 4. Provides a mechanism for inheriting slots from abstract frames, such as the disaster frame 5. None of the above 6 Reification is 1. The attempt to develop a universal representation 2. The tendency to attribute magical powers to particular mechanisms 3. The process by which ways of thinking are determined by macro and micro cultures 4. The process of using perceptions to answer questions too hard for rule-based systems 5. None of the above 7 Arch learning includes 1. A demonstration of how to combine the benefits of neural nets and genetic algorithms 2. A commitment to bulldozer computing using 100's of examples to learn concepts 3. The near miss concept 4. A central role for the Goldilocks principle 5. None of the above 8 Arch learning benefits importantly from 1. An intelligent teacher 2. Exposure to all samples at the same time 3. Use of crossover 4. Sparse spaces 5. None of the above 9 Experimental evidence indicates 1. People who talk to themselves more are better at physics problems than those who talk less 2. Disoriented rats look for hidden food in random corners of a rectangular room 3. Disoriented children combine color and shape information at about the time they start walking 4. Disoriented children combine color and shape information at about the time they start counting 5. None of the above 10 Goal trees 1. Enable rule-based systems to avoid logical inconsistency 2. Enable rule-based systems answer questions about behavior 3. Are central to the subsumption architecture's ability to operate without environment models 4. Are central to the subsumption architecture's ability to cope with unreliable hardware 5. None of the above 48

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: Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe

Mega Millions Lottery Minitab Project 1 The Mega Millions game is a lottery game that is played by picking 5 balls between 1 and 52 and additional megaball between 1 and 52. The cost per game is $1. The

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These notes will serve as a guide and reminder of several features in Excel 2007 that make the use of a spreadsheet more like an interactive thinking tool. The basic features/options to be explored are:

10/1/11 Solution: Minimax with Alpha-Beta Pruning and Progressive Deepening When answering the question in Parts C.1 and C. below, assume you have already applied minimax with alpha-beta pruning and progressive

Assessment Task Exemplars Here we set out to show something of the range of tasks designed and developed by the Shell Centre for Mathematical Education for various projects in the US and the UK over the

The Five Basic Skills of Drawing by Dan Johnson A free guide for improving your drawing skills, originally published on Right Brain Rockstar http://rightbrainrockstar.com/the-five-basic-skills-of-drawing/

Which Shape? This problem gives you the chance to: identify and describe shapes use clues to solve riddles Use shapes A, B, or C to solve the riddles. A B C 1. I have 4 sides. My opposite sides are equal.

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

Graphical Presentation of Data Guidelines for Making Graphs Titles should tell the reader exactly what is graphed Remove stray lines, legends, points, and any other unintended additions by the computer

The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures